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Hilb10Geom082910 - DECISION PROBLEMS IN EUCLIDEAN GEOMETRY...

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DECISION PROBLEMS IN EUCLIDEAN GEOMETRY Harvey M. Friedman* Ohio State University August 29, 2010 ADVANCED DRAFT Abstract. We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry, involving lines and integer points. We also consider formulations involving integral domains of characteristic 0, and ordered rings. The main tool is the solution to Hilbert's Tenth Problem. The limited number of facts used from recursion theory are isolated at the beginning. 0. PRELIMINARIES. 1. REALIZING COLINEARITY. 2. REALIZING PARALLELS. 3. REALIZING BETWEENNESS. 4. REALIZING EQUIDISTANCE. 5. ADDITIONAL RESULTS. 0. Preliminaries. We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry. The main tool is the solution to Hilbert's Tenth Problem. More generally, we demonstrate algorithmic reductions between various decision problems in two dimensional Euclidean geometry over integral domains of characteristic zero, and ordered rings, with (variants of) Hilbert's 10th Problem for such rings. All of our algorithmic reductions are uniform in the rings from the category of rings. Thus we view the reductions as algebraic or geometric, rather than combinatorial. The basic construction that drives the result is a reduction of ring addition and ring multiplication in any integral domain of characteristic zero, to collinear relations, through Lemmas 1.3 and 1.5. For addition, the realizations of the collinear relations are unique as stated in Lemma 1.3. However, for multiplication, the realizations are not unique (see Lemma 1.5).
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We begin with a brief account of the relevant background from recursion theory. It is convenient to work within a fixed "universal" space that is rich enough to naturally support the kind of finitary objects that our algorithms operate on. Accordingly, we use the least set T that contains the integers and the 52 lower case and upper case alphabetic characters, and where every finite sequence from T (possibly empty) is an element of T. We use the basic notion of partial recursive f:T T from recursion theory. Informally, this is a partial function from T into T such that the following holds. There exists an algorithm such that at each input x T, if f(x) = y then the algorithm yields the output y; if f(x) is undefined, then the algorithm yields no output. A recursive f:T T is a partial recursive f:T T whose domain is T. A recursive subset of T is a subset of T whose characteristic function is recursive. An r.e. (recursively enumerable) subset of T is the domain of a partial recursive f:T T. Let A,B T. We say that A is reducible to B if and only if there is a recursive f:T T such that for all x T, x A f(x) B. This is written A B. We write A B if and only if B A.
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